Approximate Hotspots of Orthogonal Trajectories

10/14/2017

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by   Ali Gholami Rudi, et al.

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In this paper we study the problem of finding hotspots of polygonal two-dimensional trajectories, i.e. regions in which a moving entity has spent a significant amount of time. The fastest optimal algorithm, due to Gudmundsson, van Kreveld, and Staals (2013), finds an axis-parallel square hotspot of fixed side length in O(n^2). We present an approximation algorithm with the time complexity O(n n) and approximation factor 1/4 for orthogonal trajectories, in which the entity moves in a direction parallel either to the x or to the y-axis. We also present a 1/4-approximation algorithm for finding axis-parallel cube hotspots of fixed side length for orthogonal three-dimensional trajectories.